Vector Algebra & Three Dimensional Geometry: JEE Main Previous Year Questions (2021-2026)

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JEE Main Previous Year Questions (2021-2026): 
Vector Algebra and 3D Geometry  
 
(January 2026) 
 
 
 
 
Vector Algebra 
Q1: Let P be a point in the plane of the vectors 
 such that P is equidistant from 
the lines AB and AC. If  then the area of the triangle ABP is : 
(a) 3/2 
(b)  
(c)  
(d) 2 
Ans: (b) 
Sol: 
Point P is equidistant from the lines AB and AC. Geometrically, this means point P must lie on 
the angle bisector of the angle formed by vectors AB and AC.  
First, we find the unit vectors along AB and AC: 
 
Since the magnitudes are equal, the angle bisector v is simply the sum of the two vectors: 
 
The vector AP will be in the direction of the unit vector v ^ : 
 
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JEE Main Previous Year Questions (2021-2026): 
Vector Algebra and 3D Geometry  
 
(January 2026) 
 
 
 
 
Vector Algebra 
Q1: Let P be a point in the plane of the vectors 
 such that P is equidistant from 
the lines AB and AC. If  then the area of the triangle ABP is : 
(a) 3/2 
(b)  
(c)  
(d) 2 
Ans: (b) 
Sol: 
Point P is equidistant from the lines AB and AC. Geometrically, this means point P must lie on 
the angle bisector of the angle formed by vectors AB and AC.  
First, we find the unit vectors along AB and AC: 
 
Since the magnitudes are equal, the angle bisector v is simply the sum of the two vectors: 
 
The vector AP will be in the direction of the unit vector v ^ : 
 
Determine Vector AP 
 
Calculate the Area of Triangle ABP 
The area of a triangle formed by two vectors AB and AP is given by the formula: 
 
First, compute the cross product AB × AP: 
 
 
 
Finally, the area of the triangle is: 
 
 
Q2: For three unit vectors  satisfying 
 
the positive value of k is : 
(a) 4 
(b) 5 
(c) 6 
(d) 3 
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JEE Main Previous Year Questions (2021-2026): 
Vector Algebra and 3D Geometry  
 
(January 2026) 
 
 
 
 
Vector Algebra 
Q1: Let P be a point in the plane of the vectors 
 such that P is equidistant from 
the lines AB and AC. If  then the area of the triangle ABP is : 
(a) 3/2 
(b)  
(c)  
(d) 2 
Ans: (b) 
Sol: 
Point P is equidistant from the lines AB and AC. Geometrically, this means point P must lie on 
the angle bisector of the angle formed by vectors AB and AC.  
First, we find the unit vectors along AB and AC: 
 
Since the magnitudes are equal, the angle bisector v is simply the sum of the two vectors: 
 
The vector AP will be in the direction of the unit vector v ^ : 
 
Determine Vector AP 
 
Calculate the Area of Triangle ABP 
The area of a triangle formed by two vectors AB and AP is given by the formula: 
 
First, compute the cross product AB × AP: 
 
 
 
Finally, the area of the triangle is: 
 
 
Q2: For three unit vectors  satisfying 
 
the positive value of k is : 
(a) 4 
(b) 5 
(c) 6 
(d) 3 
Ans: (b) 
Sol: 
 
also, 
 
So 
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JEE Main Previous Year Questions (2021-2026): 
Vector Algebra and 3D Geometry  
 
(January 2026) 
 
 
 
 
Vector Algebra 
Q1: Let P be a point in the plane of the vectors 
 such that P is equidistant from 
the lines AB and AC. If  then the area of the triangle ABP is : 
(a) 3/2 
(b)  
(c)  
(d) 2 
Ans: (b) 
Sol: 
Point P is equidistant from the lines AB and AC. Geometrically, this means point P must lie on 
the angle bisector of the angle formed by vectors AB and AC.  
First, we find the unit vectors along AB and AC: 
 
Since the magnitudes are equal, the angle bisector v is simply the sum of the two vectors: 
 
The vector AP will be in the direction of the unit vector v ^ : 
 
Determine Vector AP 
 
Calculate the Area of Triangle ABP 
The area of a triangle formed by two vectors AB and AP is given by the formula: 
 
First, compute the cross product AB × AP: 
 
 
 
Finally, the area of the triangle is: 
 
 
Q2: For three unit vectors  satisfying 
 
the positive value of k is : 
(a) 4 
(b) 5 
(c) 6 
(d) 3 
Ans: (b) 
Sol: 
 
also, 
 
So 
 
It is given that 
 
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JEE Main Previous Year Questions (2021-2026): 
Vector Algebra and 3D Geometry  
 
(January 2026) 
 
 
 
 
Vector Algebra 
Q1: Let P be a point in the plane of the vectors 
 such that P is equidistant from 
the lines AB and AC. If  then the area of the triangle ABP is : 
(a) 3/2 
(b)  
(c)  
(d) 2 
Ans: (b) 
Sol: 
Point P is equidistant from the lines AB and AC. Geometrically, this means point P must lie on 
the angle bisector of the angle formed by vectors AB and AC.  
First, we find the unit vectors along AB and AC: 
 
Since the magnitudes are equal, the angle bisector v is simply the sum of the two vectors: 
 
The vector AP will be in the direction of the unit vector v ^ : 
 
Determine Vector AP 
 
Calculate the Area of Triangle ABP 
The area of a triangle formed by two vectors AB and AP is given by the formula: 
 
First, compute the cross product AB × AP: 
 
 
 
Finally, the area of the triangle is: 
 
 
Q2: For three unit vectors  satisfying 
 
the positive value of k is : 
(a) 4 
(b) 5 
(c) 6 
(d) 3 
Ans: (b) 
Sol: 
 
also, 
 
So 
 
It is given that 
 
 
 
 
Q3: Let  Let  be 
the vector in the plane of the vectors  such that the length of its projection on 
the vector  Then  is equal to 
(a)  
(b) 13 
(c)  
(d) 7 
Ans: (a) 
Sol: 
Given 
 
v is in the plane of vectors a and b, 
 
It is given that length of projection of v on c is